To Stitch or Not to Stitch, That Is the Question: Multi-Gaze Eye Topography Stitching Versus Single-Shot Profilometry

Abstract

Purpose: To evaluate whether corneal topography map stitching can fully substitute the traditional single-shot capture methods in clinical settings. Methods: This record review study involved the measurement of corneal surfaces from 38 healthy subjects using two instruments: the Medmont Meridia, which employs a stitching composite topography method, and the Eye Surface Profiler (ESP), a single-shot measurement device. Data were processed separately for right and left eyes, with multiple gaze directions captured by the Medmont device. Surface registration and geometric transformation estimation, including neighbouring cubic interpolation, were applied to assess the accuracy of stitched maps compared to single-shot measurements. Results: The study evaluated eye rotation angles and surface alignment between Medmont topography across various gaze directions and ESP scans. Close eye rotations were found in the right-gaze, left-gaze and up-gaze directions, with rotation angles of around 8°; however, the down-gaze angle was around 15°, almost twice other gaze rotation angles. Root mean squared error (RMSE) analysis revealed notable discrepancies, particularly in the right-, left-, and up-gaze directions, with errors reaching up to 98 µm compared to ESP scans. Additionally, significance analyses showed that surface area ratios highlighted considerable differences, especially in the up-gaze direction, where discrepancies reached 70% for both right and left eyes. Conclusions: Despite potential benefits, the findings highlight a significant mismatch between stitched and single-shot measured surfaces due to digital processing artefacts. Findings suggest that stitching techniques, in their current form, are not yet ready to substitute single-shot topography measurements fully. Although stitching helps fit large-diameter contact lenses, care should be taken regarding the central area, especially if utilising the stitched data for optimising optics or wavefront analysis.

1. Introduction

Eye care practitioners (ECPs) traditionally relied on keratometry to evaluate corneal shape, measuring a 3 to 4 mm annulus on the central cornea [1]. While suitable for fitting small rigid corneal lenses, keratometry is less effective for larger diameter lenses that extend towards or beyond the limbus. Over the past three decades, computerised topography has enhanced understanding of corneal shape by projecting concentric ring patterns onto the anterior tear film, generating topographic maps [2]. However, most maps cover only about 9 mm horizontally and 7.5 mm vertically [3], often excluding the limbal region critical for fitting soft lenses [4,5,6]. Although scanning tomography offers additional anterior segment data, it is limited by the number of scanned meridians (e.g., 25 or 50 for Pentacam devices).

Optical fringe projection technology has led to the development of numerous profilometry methods, including Fourier transform profilometry (FTP) [7], phase-shifting profilometry (PSP) [8], Moiré profilometry (MP) [9], computer-generated Moiré profilometry (CGMP) [10], modulation measuring profilometry (MMP) [11], and phase-differencing profilometry (PDP) [12]. These techniques have substantially advanced the field of surface measurement, including applications in ophthalmic imaging. The Eye Surface Profiler (ESP), developed in 1998, provides a three-dimensional (3D) height map of up to 20 mm of the ocular surface using FTP. This method incorporates two blue light projectors, a yellow filter camera, and fluorescein dye to capture over 250,000 data points [13]. The ESP aids lens fitting and visualising the cornea–sclera transition [14]. Placido-based instruments, like the Medmont Meridia (Medmont International Pty Ltd., Nunawading, Australia), have extended their measurement range through map stitching, combining data from multiple gaze directions [15]. The sMap3D (Precision Ocular Metrology) also uses stitching to create maps. However, comparisons of these stitched maps with single-shot ESP maps remain limited [15,16].

Franklin et al. explored reverse engineering Medmont Meridia stitching using axial curvature maps, which assumes the radii of curvature centres align with the optical axis, a limitation in representing the true ocular surface [17]. Given the proprietary nature of commercial signal processing and known limitations in eye scanning systems [18,19,20], a reverse engineering approach is used here to evaluate the Medmont Meridia’s stitching against the single-shot ESP instrument.

However, multi-gaze stitching offers broader surface coverage at a lower cost than single-shot profilometry, raising concerns about accuracy. This study investigates whether the stitching of Placido-based maps, with all the individual patient variability of gaze fixation direction and fixation accuracy, can be as accurate as a measurement taken by a single-shot instrument where the patient does not move their eyes. It assesses the readiness of stitching technology in its current form to fully substitute single-shot topography measurements.

2. Materials and Methods

2.1. Clinical Data Collection

This record review study was approved by the ethical committee of the Institutional Review Board in Taiwan (CS1-23215) and conducted following the standards set in the Declaration of Helsinki. The study was based solely on existing anonymised data with no new recruitments, where the eyes of 38 healthy subjects with no previous ophthalmic surgeries were measured by Medmont Meridia using the Composite Topography Wizard and then by the ESP in a single-shot measurement each on the same day and all by the same operator. The inclusion criterion required participants to have no ocular diseases except for ametropia refractive errors. Exclusion criteria included any history of eye surgery, presence of ocular surface disease or scarring, diagnosis of connective tissue disease, and being pregnant or in the early postpartum period.

2.1.1. Medmont Meridia

To capture multiple images in Medmont Meridia software, the Composite Topography Wizard guides the operator to capture five images (Figure 1). In automatic mode, the instrument captures the straight-ahead gaze and then directs the examiner to ask the subject to gaze up, down, left and right. Images are automatically taken when the pupil reaches the required offset from the axis.

Suppose the participant’s pupil cannot be detected or they are unable to move their gaze sufficiently to trigger auto-capture. The examiner can manually click the “Next” button to capture images using the current fixation location. The composite registration algorithm utilises the pupil’s location as a starting point, so the pupil must be accurately identified in each image. However, only measurements successfully taken in automatic mode were considered in this study.

2.1.2. ESP

For the ESP, a wetting agent using sodium hyaluronate was instilled into the eye before pre-aligning the device, after which fluorescein was instilled into the eye. The lids were required to be separated to allow sufficient access to the ocular surface, and the subject was instructed to fixate on a red cross. The device was then re-aligned, and three measurements were taken in succession. For the purpose of this study, the three processed maps were assessed for the amount and evenness of coverage, and the best one was selected by the same operator and numerically compared to the Medmont Meridia stitched maps via custom-built software. While the best of each of the three ESP scans was chosen for the analysis, all three scans were used to test the repeatability of ESP scans by calculating the RMSE among them.

2.1.3. Processing the Instruments’ Output

Right and left eyes were processed separately to avoid confounding factors appearing because of using fellow eyes in the same set of analyses [21]. There was also a technical reason for this separation, as the right-gaze for the right eye is towards the nasal side, while the right-gaze for the left eye is towards the temporal side and vice versa. In addition, flipping a left eye surface to be analysed alongside a right eye surface was considered an unsuitable choice as the superior–inferior line was not a proper mirror line when the eye gaze was in non-axial directions [22]. Data were exported (6 files per eye) from the Medmont Meridia in extensible markup language (xml) format as xmf files and (three files per eye) from the ESP in MATLAB (The MathWorks Inc., Natick, MA, USA) mat format. As the ESP has some marginal noises in the far periphery, the so-called edge effect, the artefact-free method developed in [19] was applied to ensure an artefact-free surface was processed. Medmont Meridia files corresponded to stitched composite (combined), ahead, right-, left-, up-, and down-gaze corneal topography, respectively.

2.2. Geometric Transformation Estimation

The process of aligning two or more three-dimensional (3D) surfaces to match is known as surface registration. Topography stitching in the context of this study refers to surface registration mathematical techniques used to broaden scan coverage area through decent orientation and integration, ensuring that corresponding points match as closely as possible. This process produces what is so-called composite (stitched) corneal topography in the Medmont software.

The traditional iterative closest point (ICP) was initially considered for surface registration due to its previous effective use in the literature [22,23]. However, ICP was disregarded because of its relatively high RMSE due to its sensitivity to initial alignment and convergence to local minima [24], especially with digital noise and edge outliers in place [25]. Additionally, it was computationally expensive and time-consuming when the number of iterations sat at 20 or more to improve its convergence.

Instead, the current study used an adapted neighbouring cubic interpolation (NCI)-based geometric transformation estimation algorithm to estimate geometric transformations in 3D space between two sets of 3D point clouds [26,27]. With the surfaces that need to be matched being constructed by different points and XY-grids, the method starts with using NCI to fit both surfaces to a wide range XY-grid that contains the visible solution of the moving surface while matching a datum surface. NCI requires at least four points in each dimension and a uniform spacing that could vary among dimensions but with a constant number of points for each. Fortunately, this limitation did not affect this study, as the number of Medmont Meridia measured points was very well above 4, and spacing was always uniform. By applying NCI, both surfaces had the same number of points, while Z values of the XY-grid points with no measured height were set to a scalar representation of “not a number” known as NaN. Hence, the transformation process was triggered.

The term transformations in this framework indicates the combination of rotation and translation. It maps a set of 3D moving points ($\mathrm{M}$), consisting of N points, to a 3D fixed set of points that work as a datum ($\mathrm{D}$) through a rotation matrix and translation vector. Given two sets of corresponding 3D points, $\mathrm{D}=\left\{{\mathrm{d}}_1,{\mathrm{d}}_2,\dots ,{\mathrm{d}}_{\mathrm{n}}, \dots ,{ \mathrm{d}}_{\mathrm{N}}\right\}$ and $\mathrm{M}=\left\{{\mathrm{m}}_1,{\mathrm{m}}_2,\dots ,{\mathrm{m}}_{\mathrm{n}}, \dots ,{ \mathrm{m}}_{\mathrm{N}}\right\}$, where each ${\mathrm{d}}_{\mathrm{n}}$ and ${\mathrm{m}}_{\mathrm{n}}$ represent points in the 3D space, the objective was to find a rotation matrix $\mathrm{R}$ and a translation vector $\mathrm{T}$ such that the points in $\mathrm{M}$ are transformed to align with $\mathrm{D}$ as closely as possible. Mathematically, the anterior eye was presented in a standard 3D coordinate system where X was the nasal–temporal axis, Y was the superior–inferior axis, and Z was the axial axis of the eye (Figure 2).

Therefore, the problem was formulated as an optimisation exercise that finds $\mathrm{R}$ and $\mathrm{T}$ that minimise the sum of squared distances between the transformed points of $\mathrm{M}$, called ${\mathrm{M}}_{\mathrm{tr}}$, and the corresponding fixed points in $\mathrm{D}$, as described in Equation (1).

$$\underset{\mathrm{R},\mathrm{T}}{\min }\Vert {\mathrm{M}}_{\mathrm{t}\mathrm{r}}-\mathrm{D}{\Vert }^2$$

(1)

where transformed points ${\mathrm{M}}_{\mathrm{tr}}$ are obtained from $\mathrm{M}$ by applying the rotation and translation matrices, then the minimisation objective function is expressed as in Equation (2).

$$\underset{\mathrm{R},\mathrm{T}}{\min }\Vert \mathrm{R}\ast \mathrm{M}+\mathrm{T}-\mathrm{D}{\Vert }^2$$

(2)

The rotation matrix $\mathrm{R}$ must be orthogonal, i.e., ${\mathrm{R}}^{\mathrm{T}}\mathrm{R}=\mathrm{I}$, where $\mathrm{I}$ is the identity matrix, and has a determinant of one to produce a valid 3D rotation. The $\mathrm{R}$ matrix that resulted from the geometric transformation estimation algorithm can be expressed as seen in Equation (3).

$$R=\left[\begin{array}{ccc}\cos {\mathrm{\alpha }}_{\mathrm{x}}\cos {\mathrm{\alpha }}_{\mathrm{y}}& \cos {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{y}}\sin {\mathrm{\alpha }}_{\mathrm{z}}-\sin {\mathrm{\alpha }}_{\mathrm{x}}\cos {\mathrm{\alpha }}_{\mathrm{z}}& \cos {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{y}}\cos {\mathrm{\alpha }}_{\mathrm{z}}+\sin {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{z}}\\ \sin {\mathrm{\alpha }}_{\mathrm{x}}\cos {\mathrm{\alpha }}_{\mathrm{y}}& \sin {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{y}}\sin {\mathrm{\alpha }}_{\mathrm{z}}+\cos {\mathrm{\alpha }}_{\mathrm{x}}\cos {\mathrm{\alpha }}_{\mathrm{z}}& \sin {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{y}}\cos {\mathrm{\alpha }}_{\mathrm{z}}-\cos {\mathrm{\alpha }}_{\mathrm{x}}\sin {\mathrm{\alpha }}_{\mathrm{z}}\\ -\sin {\mathrm{\alpha }}_{\mathrm{y}}& \cos {\mathrm{\alpha }}_{\mathrm{y}}\sin {\mathrm{\alpha }}_{\mathrm{z}}& \cos {\mathrm{\alpha }}_{\mathrm{y}}\cos {\mathrm{\alpha }}_{\mathrm{z}}\end{array}\right]$$

(3)

where ${\mathsf{\alpha }}_{\mathrm{x}}$ is the rotation angle around the X-axis, ${\mathsf{\alpha }}_{\mathrm{y}}$ is the rotation angle around the Y-axis, and ${\mathsf{\alpha }}_{\mathrm{z}}$ is the rotation angle around the Z-axis. Likewise, the translation vector $\mathrm{T}$ can be expressed as seen in Equation (4).

$$\mathrm{T}=\left[\begin{array}{c}{\mathrm{X}}_{\mathrm{t}}\\ {\mathrm{Y}}_{\mathrm{t}}\\ {\mathrm{Z}}_{\mathrm{t}}\end{array}\right]$$

(4)

where ${\mathrm{X}}_{\mathrm{t}},{ \mathrm{Y}}_{\mathrm{t}}$, and ${\mathrm{Z}}_{\mathrm{t}}$ are the translations in X, Y, and Z directions, respectively. Hence, the transformed anterior eye coordinate can be expressed as shown in Equation (5).

$$\left[\begin{array}{ccccc}{\mathrm{x}}_{\mathrm{t}\mathrm{r} 1}& {\mathrm{x}}_{\mathrm{t}\mathrm{r} 2}& {\mathrm{x}}_{\mathrm{t}\mathrm{r} 3}& \dots & {\mathrm{x}}_{\mathrm{t}\mathrm{r} \mathrm{N}}\\ {\mathrm{y}}_{\mathrm{t}\mathrm{r} 1}& {\mathrm{y}}_{\mathrm{t}\mathrm{r} 2}& {\mathrm{y}}_{\mathrm{t}\mathrm{r} 3}& \dots & {\mathrm{y}}_{\mathrm{t}\mathrm{r} \mathrm{N}}\\ {\mathrm{z}}_{\mathrm{t}\mathrm{r} 1}& {\mathrm{z}}_{\mathrm{t}\mathrm{r} 2}& {\mathrm{z}}_{\mathrm{t}\mathrm{r} 3}& \dots & {\mathrm{z}}_{\mathrm{t}\mathrm{r} \mathrm{N}}\end{array}\right]=\mathrm{R}\ast \left[\begin{array}{ccccc}{\mathrm{x}}_{\mathrm{m} 1}& {\mathrm{x}}_{\mathrm{m} 2}& {\mathrm{x}}_{\mathrm{m} 3}& \dots & {\mathrm{x}}_{\mathrm{m} \mathrm{N}}\\ {\mathrm{y}}_{\mathrm{m} 1}& {\mathrm{y}}_{\mathrm{m} 2}& {\mathrm{y}}_{\mathrm{m} 3}& \dots & {\mathrm{y}}_{\mathrm{m} \mathrm{N}}\\ {\mathrm{z}}_{\mathrm{m} 1}& {\mathrm{z}}_{\mathrm{m} 2}& {\mathrm{z}}_{\mathrm{m} 3}& \dots & {\mathrm{z}}_{\mathrm{m} \mathrm{N}}\end{array}\right]+\mathrm{T}$$

(5)

RMSE between the moving (rotated and translated) surface and fixed surfaces was recalculated, as shown in Equation (6).

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{{\displaystyle \sum }}_{\mathrm{i}=\mathrm{1}}^{\mathrm{N}}\left({\left({\mathrm{x}}_{\mathrm{t}\mathrm{r} \mathrm{i}}-{\mathrm{x}}_{\mathrm{d} \mathrm{i}}\right)}^2+{\left({\mathrm{y}}_{\mathrm{t}\mathrm{r} \mathrm{i}}-{\mathrm{y}}_{\mathrm{d} \mathrm{i}}\right)}^2+{\left({\mathrm{z}}_{\mathrm{t}\mathrm{r} \mathrm{i}}-{\mathrm{z}}_{\mathrm{d} \mathrm{i}}\right)}^2\right)}{\mathrm{N}}}}$$

(6)

where the subscript $\mathrm{tr}$ stands for the surface that has been transformed to match the datum surface, subscripted as $\mathrm{d}$; while subscript $\mathrm{i}$ indicates individual points and $\mathrm{N}$ is the number of points (Figure 3).

The algorithm checks for a transformation from a moving surface to a datum surface and then calculates the distance between the matched points in each pair. If the distance between the matched points in a pair was greater than the maximum distance value, which was set to 0.1 mm, then the pair was considered an outlier for that transformation. The pair was regarded as an inlier if the distance was less than the specified maximum. The function excluded outliers using the MATLAB-impeded M-estimator sample consensus (MSAC) algorithm that extended the random sample consensus algorithm by using a cost function that penalised outliers based on their residual errors, enhancing robustness in noisy data environments. This approach ensured more accurate matching by better distinguishing between inliers and outliers.

2.3. Testing Geometric Transformation Estimation as a Reverse Engineering Approach

A testing method was developed to test the validity of geometric transformation estimation as a reverse engineering approach in stitching. Each ESP-measured surface was used individually. A random portion of the surface, within a 7 mm diameter circle centred at a random location, was selected. This portion was then translated and rotated by random displacements (ranging from −3 to 3 mm) and angles (ranging from −15° to 15°) using MATLAB’s rand function. After these transformations, the geometric transformation estimation method described in this study was applied to reverse these changes and return the portion as close as possible to its original position. The RMSE was then calculated. The numerical ranges for this test were selected based on a preliminary study, which identified them as appropriate considering the anterior eye surface area covered by the ESP.

2.4. Statistical Analyses

In this study, the sample size justification leverages the paired design, where each of the 38 subjects was measured by two instruments, enhancing the efficiency of analysing differences. Cohen’s effect size was calculated based on the mean differences between paired measurements, and the larger sample size further reduces variability and improves the precision of the estimates. Considering a moderate effect size, bigger than 0.5, and aiming for a power of 80% at a 5% significance level, this sample size ensures robust sensitivity while supporting the generalisability of the results. While a smaller sample of 17 paired measurements can statistically detect differences with adequate sensitivity, increasing the sample size to 38 further reduces variability and improves the precision of estimates. The expanded sample size balances practicality with statistical rigour, making results suitable for confirming findings across measurements in the current exploratory comparison.

The statistical analysis was conducted using MATLAB Statistics and Machine Learning Toolbox, explicitly utilising the Kruskal–Wallis test to investigate differences between the corneal surface measurements obtained from the Medmont Meridia, which employs a stitching technique, and the ESP, which uses a single-shot measurement. The Kruskal–Wallis test was applied to RMSE and rotational misalignment angles across different gaze directions between the two measurement methods. A significance level of p < 0.05 was set to determine statistical significance [28]. When the test indicated significant differences, post hoc pairwise comparisons were performed to identify specific contrasts between the two measurement groups.

The p-value results were presented in maps that compared relevant surface areas. This representation provides insights into surface topography measurement variability, highlighting differences between the stitching and single-shot methods. The study only compared two groups at a time, ESP versus a gaze direction, so the multiple comparison problems did not apply as a single statistical test was conducted one after another. Therefore, there was no need for a correction like the Bonferroni correction, as there was no risk of inflated false positives from multiple comparisons.

3. Results

This section represents three sets of results, respectively. Firstly, the five reverse-engineered Medmont Meridia topographies were compared to the stitched surface. The process allowed for determining the angles of eye gaze in each direction. These angles represented the eye rotation around the y-axis for the right- and left-gaze and around the x-axis for the up- and down-gaze. The second set of results reported the same angles when the Medmont-scanned surface was registered against the ESP-scanned surface for the same subjects. Finally, a comparison of the RMSE between the two sets of registration processes was made to determine the quality of the stitching process in each case.

For both the right and left eyes, Figure 4 and Figure 5 show the eye rotation angles during Medmont Meridia topography stitching, compared to the combined surface. In the straight-ahead gaze direction, the deviations were minimal, such as α_x = 0.00° for the right eye and −0.02° for the left eye. However, noteworthy rotations in the right- and left-gaze directions can be seen, especially in the α_y angle, where the right eye shows 8.06° and the left eye −8.22°. On up-gaze, there was a notable rotation in α_x, with the right and the left eyes at 7.62°. The down direction showed the largest rotations in α_x, −15.12° for the right eye and −15.17° for the left eye.

Although only the best scan from each set of three ESP scans was selected for analysis, all scans were utilised to assess repeatability by calculating the RMSE. The ESP RMSE values were 4.23 ± 4.15 µm for right eyes and 3.65 ± 2.62 µm for left eyes.

Moving to register Medmont Meridia multiple gaze scans to the ESP single-shot scan as a datum, another set of eye rotation angles was found. Hypothetically, if the two machines gave the exact measurements, the new set of angles should be identical to the previous one. Figure 6 and Figure 7 present the eye rotation angles during Medmont Meridia topography stitching, compared to ESP single-shot measurements for both eyes. The combined surface displayed only slight rotations, with values close to zero in all directions. Minimal rotations were observed in the straight-ahead gaze. In contrast, noteworthy rotations occurred in the right- and left-gaze directions, especially in the α_y angle, reaching 8.35° for the right eye and −8.72° for the left eye. The largest rotations were found in the down-gaze, with the α_x angle showing −15.44° for the right eye and −15.45° for the left eye.

Figure 8 and Figure 9 display the RMSE for Medmont Meridia scans compared to the combined surface and ESP scans for both eyes. The combined surface had no error when Medmont scans were compared to themselves, but noteworthy errors appeared when compared to ESP scans, with RMSE values of 57 µm for the right eye and 53 µm for the left. In the straight-ahead gaze, Medmont scans had low errors (around 4 µm), while ESP scans had higher errors, reaching 23 µm for the right eye and 27 µm for the left. Larger errors were observed in the right and left-gaze directions, with Medmont scans showing RMSEs of up to 49 µm, and ESP scans having even higher errors, reaching 98 µm.

Similarly, in the left-gaze direction, Medmont scans showed errors of 49 µm for the right eye and 39 µm for the left, while ESP scans had larger errors of 98 µm and 63 µm, respectively. When gazing up, Medmont scans had RMSEs of 37 µm for both eyes, but ESP scans had higher errors, with 93 µm for the right eye and 82 µm for the left. In the down-gaze, Medmont errors were smaller, at 15 µm for the right eye and 16 µm for the left, compared to ESP errors of 40 µm for both eyes.

Figure 10 and Figure 11 reveal significant differences in surface area ratios between Medmont and ESP scans for both eyes, where lower ratios indicate a better match. The combined-gaze fixation showed large disparities, with ratios of 59% for the right eye and 26% for the left. In contrast, the ahead-gaze direction had the smallest differences, with a nearly perfect match at 0% for the right eye and 1% for the left. The right-gaze showed notable differences, with ratios of 62% for the right eye and 61% for the left. Similarly, the left-gaze had 61% and 60% ratios. The most significant differences were found in the up-gaze, with 70% for both the right and left eyes.

Regarding testing geometric transformation estimation as a reverse engineering approach, the test recorded a mean RMSE of 0.88 ± 0.34 μm for the right eyes and 0.93 ± 0.39 μm for the left eyes. Compared to the minimum RMSE values recorded in the current study (approximately 4 μm), these validation RMSE values were considered negligible (less than 0.25% of the minimum recorded value). This demonstrates the reliability of the technique.

Additional numerical and graphical results that support the findings of this study are provided in the Supplementary Materials. These supplementary sections include detailed quantitative analyses, such as root mean square error (RMSE) values across different gaze directions, eye rotation angle measurements, and statistical comparisons using post hoc and Kruskal-Wallis tests (see Supplementary Section S1). Furthermore, graphical representations of surface registration accuracy and rotation data are presented in Sections S2 to S4, offering visual context to the alignment process and the consistency of eye movement patterns. Together, these materials provide a comprehensive view of the data and methodology, enhancing the transparency and reproducibility of the study.

4. Discussion

In the realm of ophthalmic imaging, the quest for precision in mapping the corneal surface has led to the widespread adoption of topography stitching techniques. This approach, which involves capturing multiple eye scans and integrating them into a unified topographic map, has been instrumental in advancing the understanding of corneal curvature and surface irregularities in peripheral areas of the eye. However, this method has limitations concerning accuracy, efficiency, and patient comfort with multiple consecutive measurements.

Topography stitching relies heavily on surface registration, a complex process that involves aligning and merging multiple shots to create a coherent representation of the corneal surface.

Despite advancements in signal processing algorithms, this technique is inherently prone to errors such as misalignment, edge artefacts, and inconsistencies in data integration. These issues can compromise the overall accuracy of the topographic map, potentially affecting diagnostic outcomes and treatment planning. After all, both Medmont Meridia and ESP measure the reflection from the tear layer, not the ocular surface itself; therefore, variations in the tear layer during multiple shots are inevitable and cannot be avoided.

In contrast, single-shot profilometry presents a compelling alternative. Single-shot techniques eliminate the need for multiple captures and complex stitching processes by capturing the surface profile in one image. This approach not only enhances accuracy by reducing potential sources of error associated with image registration but also improves efficiency and patient comfort by minimising the time required for image acquisition.

Unlike the traditional tendency of iteratively fitting one or both of the moving (M) and datum (D) sets to circular domain functions, like Zernike polynomials [29], the current study algorithm uses NCI once instead. NCI outperforms Zernike polynomials in interpolation tasks due to its local fitting capability, simplicity, and flexibility in handling non-uniformly spaced data. While NCI provides a smooth, continuous curve that adapts well to regional variations, Zernike polynomials offer a global fit considering all points at once and are more suited to concentric circular domains, such as optical wavefront analysis targeting the centre of the cornea. For non-optical applications, such as registering rotated, shifted and misaligned surfaces, the effectiveness of NCI makes it a preferred choice considering the known limitations of Zernike polynomials [30,31].

The current study considered ESP, which uses optical fringe projection technology, as a single-shot anterior eye measuring system. Optical fringe projection has revolutionised surface profilometry by enabling high-precision, non-contact measurement of complex surfaces. This technology works by projecting structured light patterns onto an object and analysing the resulting deformations to reconstruct its 3D geometry. Several profilometry techniques have emerged from this approach, each with unique principles and applications. FTP extracts surface height information by applying Fourier analysis to a single deformed fringe pattern, making it efficient for active measurements [32]. PSP enhances accuracy by capturing multiple phase-shifted images and computing phase differences for sub-micron precision [33]. MP [34] and CGMP [10] utilise interference patterns to detect surface variations, which are commonly used in biomedical imaging. MMP introduces intensity modulation to improve measurement sensitivity [11], while PDP refines phase extraction for enhanced robustness against noise [12]. These techniques have significantly advanced surface measurement capabilities, particularly in ophthalmic imaging, where high-resolution corneal and scleral topography is crucial for diagnostics, contact lens fitting, and refractive surgery planning.

Existing single-shot profilometry methods include techniques such as RGB compound colour profilometry using wavelength multiplexing, where different colours encode phase information simultaneously, allowing instant 3D reconstruction [35]. Orthogonal modulated profilometry, on the other hand, employs spatially encoded fringe patterns with distinct orthogonal modulations to recover depth without requiring multiple phase shifts [36]. These methods enhance real-time surface measurement, making them particularly effective in dynamic applications such as biomedical imaging and corneal topography.

A critical aspect of fringe projection-based profilometry is the phase unwrapping process, which resolves the inherent phase ambiguity caused by the arctangent operation [33]. The accuracy of phase unwrapping influences the reliability of the reconstructed surface [32]. While the current study primarily referenced hierarchical multi-frequency phase unwrapping, a broader range of spatial phase unwrapping (SPU) [37] and temporal phase unwrapping (TPU) [38] techniques exist, each with distinct advantages and limitations [39].

SPU methods process the unwrapped phase within a single frame, making them suitable for static or high-resolution imaging applications [40]. Among these, quality-guided SPU follows a high-to-low reliability gradient, minimising error propagation [41]. The branch-cut SPU method addresses phase discontinuities by strategically placing branch cuts to prevent inconsistencies [38]. Advanced techniques, such as rhombus-type SPU [42], curtain-type SPU [43], and multi-anchors bidirectional suppression SPU [44], further refine the correction process by integrating directional phase continuity constraints, enhancing their utility in complex or irregular surface reconstructions [45].

In contrast, TPU methods rely on multiple structured patterns or phase shifts over time to extract absolute phase information, reducing spatial discontinuities and noise sensitivity [38]. Techniques such as grey code TPU and heterodyne multi-frequency TPU improve accuracy by encoding fringe patterns with distinct phase information across frames [46]. Hierarchical multi-frequency TPU enhances robustness by progressively refining phase estimates [47]. Additional approaches, including fringe amplitude encoding TPU [48], binary coding TPU [49], phase shift coding TPU [48], and phase shift coding division multiplexing TPU [50], incorporate various modulation and encoding strategies to further mitigate phase ambiguity in challenging measurement conditions [41].

Phase-to-height mapping is fundamental in profilometry, converting the measured phase information into absolute height data for accurate 3D surface reconstruction. The accuracy of this mapping process influences the precision of profilometry-based measurements, making it essential to integrate robust phase-to-height algorithms [51]. Traditional approaches rely on system calibration and geometric transformation models, where the phase shift is mapped to height using known optical parameters. Recent advancements, however, have introduced enhanced phase-to-height mapping methods, including device-calibrated mapping, direct phase-to-height transformations, and multi-plane phase reference techniques, which aim to reduce systematic errors, improve measurement speed, and enhance the accuracy of reconstructed surfaces [52].

Xiao et al. proposed an improved algorithm for phase-to-height mapping, where both phase measurement and system calibration are completed simultaneously, thereby enhancing the efficiency and accuracy of three-dimensional measurements [51]. Additionally, integrating device calibration with phase-to-height mapping has been shown to minimise environmental distortions and optical misalignments, significantly improving measurement precision [52]. Furthermore, Zhang and Yau explored direct phase-to-height mapping approaches, eliminating the need for geometric parameters by using multiple parallel phase references to construct a direct height-mapping formula, simplifying the measurement process, and reducing computational complexity [53]. By incorporating these advancements, phase-to-height mapping techniques continue to evolve, offering greater precision in optical profilometry applications. Future research should focus on developing AI-driven mapping models and adaptive calibration techniques to further enhance measurement accuracy across diverse experimental conditions.

Ensuring accuracy in 3D metrology is crucial, as various error sources can impact measurement precision. Nonlinear phase errors arise due to the nonlinear response of digital projectors and cameras, leading to distortions in projected fringe patterns [38]. Correction methods such as phase error compensation models help mitigate these distortions [54]. The gamma effect of projectors introduces further inaccuracies by distorting sinusoidal fringe patterns, requiring gamma correction techniques for improved accuracy [54]. Light source stability also plays a critical role, as fluctuations in intensity affect phase consistency, making temperature-controlled LED sources a preferable solution [39].

Additionally, camera sensitivity variations, including exposure time and sensor noise, can introduce measurement errors. Strategies like multi-exposure phase retrieval and high-dynamic-range (HDR) imaging enhance robustness [38]. Addressing these error sources through correction algorithms, calibration techniques, and advanced hardware solutions is essential for improving measurement accuracy and system reliability in 3D profilometry.

With an average ESP RMSE of less than 5 µm when its repeatability was tested, the current study showed that Medmont ahead-gaze fixation recorded a 22 ± 9 µm RMSE compared to the ESP. At the same time, the stitched Medmont surface exhibited a 48 ± 15 µm RMSE, which was more than twice that of the ahead-gaze fixation. These findings indicate a notable difference in accuracy between the two scanning methods. Since both comparisons were subject to the same ratio of measurement artefacts, if any, it could be inferred that the standard ahead-gaze Medmont scan was more accurate than the stitched surface when benchmarked against ESP scans. This discrepancy highlights the potential limitations of the stitching process in Medmont scans, which may introduce additional errors do not present in single-shot measurements. The higher RMSE in the stitched surface suggests that the complexity involved in combining multiple scans may compromise the overall accuracy. Therefore, while the stitched surface can provide a more comprehensive topography, it is crucial to acknowledge the trade-offs in precision.

The study provides new map types that demonstrate where the differences between Medmont-measured individual or stitched surfaces and single-shot ESP-measured surfaces have significance. These maps show that the best match, with insignificant RMSE (p > 0.05), between the Medmont-stitched and ESP single-shot surfaces was observed in the Medmont ahead-gaze scan and the outer edges of the Medmont-stitched surface approximately beyond a 3.5 mm radius from the corneal centre, with areas adjacent to the central cornea being less precise. This indicates that although the periphery of a combined map matched a single-shot measurement, the central region was less accurate, and the standard, straight-ahead map was a better representation of this area.

Future studies should investigate innovative methods to minimise stitching errors and improve the alignment algorithms to enhance the reliability of combined surface measurements. Additionally, exploring the effective integration of ahead-gaze data with other directions could offer a balanced approach, leveraging the accuracy of single-shot scans while still capturing a broader topographical landscape. This consideration is essential for improving clinical exercise and ensuring the most accurate anterior eye surface measurements are used in practice.

5. Conclusions

As the research team has no access to the source code of the commercial software packages of Medmont Meridia or ESP, their conclusions are based on their reverse engineering process, which was necessary to investigate such inaccessible stitching algorithms that have been used in clinical settings for years.

This study revealed significant differences in accuracy between the Medmont ahead-gaze fixation when considered alone and the combined stitched surface when compared to ESP scans. The ahead-gaze fixation demonstrated a notably lower RMSE, indicating higher accuracy than the stitched surface. This suggests that while the stitched surface provides a more comprehensive topographical map regarding coverage, it may introduce additional errors in the central area. These findings highlight the importance of optimising scanning and stitching techniques to ensure precise and reliable eye measurements. Based on the obtained results for this study, the best match between the Medmont-stitched and the ESP single-shot surfaces can be found in the Medmont ahead scanned shot and the periphery of the Medmont-stitched surface beyond a 3.5 mm radius from the corneal centre.

During the reverse engineering process in the current study, Zernike polynomials were not considered as they are global surface fitters and are limited to well-centred circular domains; hence, they struggle with capturing local variations unless a high degree is selected [31]. Even though they risk instability at higher degrees and lack flexibility for irregular, misaligned or non-circular data [30].

While topography stitching extends the mapping of the ocular surface, it is clear that it has its drawbacks. The process of aligning and merging multiple images can lead to errors and inconsistencies, which might impact the accuracy of the final combined map. This may have implications for any embedded contact lens software for large-diameter contact lenses that relies on the combined surface for recommended fits or any ocular research investigating relationships between central and peripheral ocular areas. Single-shot profilometry still provides a more straightforward solution by capturing the entire surface in just one go. Single-shot technology not only boosts accuracy by sidestepping the complications of image registration but also speeds up the process and makes the experience more comfortable for patients. Given these benefits, single-shot profilometry stands out as a more efficient method for precise corneal surface assessment, offering a significant improvement over traditional topography stitching.